Great ideas and gems of mathematics

Pierre de Fermat (1601 – 1666) was a French lawyer and amateur mathematician who made numerous contributions to mathematics (number theory, geometry, optics) but who is most famous for what did most probably did not do. Around 1637 Fermat was reading the book Arithmetica by the Greek mathematician Diophantus (the father of algebra).

Diophantus was interested in finding integer solutions to equations (or systems of equations). For example, an example of a Diophantine problem is: given two integers A, B, can you find integers x and y so that A = x + y and B = x^2 + y^2? Another is: given two integers a and b, are there integers x and y so that a x + b y = gcd(a, b)?

Another Diophantine type problem is: can you find integers x, y, z so that x y z ≠ 0 (this just means that none of x, y, z can be zero) and x^2 + y^2 = z^2. We saw in our discussion of the Pythagorean theorem that there are indeed an infinite many of solutions to this problem – called Pythagorean triples. Fermat considered the following generalization this problem:

Let’s make sure we understand what we mean here. We want to find integer x, y, z what solve this equation. There will always be non-integer solutions. For example x = 1, y = 1, and z = 2^(1/n) (the nth root of 2) is a solution to x^n+ y^n = z^n but the nth root of 2 is not an integer so this solution does not qualify. Secondly we want x, y, z to all be non-zero or else the problem becomes trivial since x = 0, y = 0, z = 0 works, as does x = 0, y = 2, z = 2, etc.

Fermat was able to show that x^4 + y^4 = z^4 has no non-zero integer solutions and conjectured that for any fixed n, x^n + y^n = z^n also has no non-zero integer solutions and made wrote the following statement – which goes down as the most famous and controversial statement in the history of mathematics.

I have discovered a truly marvelous proof that it is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. This margin is too narrow to contain it.

The above image is an image the 1621 translation of Diophantus whose margin could not hold Fermat’s “marvelous proof”. Click here for a larger image – and a chance to review your high school Latin!

Did Fermat actually have a proof of this statement? Probably not. O.K. I said it — definitely not. But this problem attracted the attention of many mathematicians of his time. Following Fermat, other mathematician show that x^n + y^n = z^n does not have an integer solution for particular values of n. Here is a brief table

n = 3 (Euler 1770)n = 4 (Fermat 1637)n = 5 (Dirichlet, Legendre 1825)

Others were able to verify other values of n and this problem got to be known as Fermat’s last theorem. Let us note the we really don’t need to check that x^n + y^n = z^n does not have a solution for all integers n > 2. We just need to check it for prime exponents (greater than two). Why is this?: Suppose that x^n + y^n = z^n has an integer solution (with of course x y z ≠ 0). Then since n = m p where p is a prime (factor n into its prime factors) then the equation x^n + y^n = z^n becomes (x^m)^p + (y^m)^p = (z^m)^p. So if the first equation has integer solutions then so does the second.

Sophie Germain (1776 – 1831) made the observation that for odd primes p less than 100 (i.e., 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97) x^p + y^p = z^p has no non-zero solutions. With the use of computers in the twentieth century, mathematicians were able to verify (through several steps as the computers became faster) Fermat’s last theorem for every prime less than four million. But of course this is not enough to verify Fermat’s last theorem since there are, as we have seem with Euclid, infinite many primes. So verifying Fermat for a large number of primes does not really get you very far.

In the later part of the twentieth century mathematicians such as Frey, Serre, Ribet were able to make some progress and relate this so seemingly trivial problem – which on its surface seems almost like a silly parlor game – to many deep ideas in mathematics concerning curves and the shape of space. Fermat’s last theorem now graduated from a silly game to an absolute necessity. It was finally solved, after fixing an error, by the Princeton professor Andrew Wiles in 1993.

I’m only giving you the brief history here. The story around this problem is amazing and you are encouraged to read Simon Singh’s account of all this in Fermat’s Enigma (required reading for this course). There is also this great video.

To give the student a little bit of the idea on how these types of problems are addressed and to show just how difficult they are to prove. We will prove in class Fermat’s original result (though not his original proof) of the following:

Proof: See class notes or the following link from Pete Clark from University of Georgia.

Note that as a corollary to this theorem we see that Fermat’s last theorem is true for n = 4 k. Can you prove this. In general, if Fermat’s theorem is true for some integer n then it is also true for any integer multiple of n.

We will comment here that Andrew Wiles received many prizes for his work on solving Fermat’s last theorem, including a knighthood from Queen Elizabeth II and the Pythagorean prize.

Finally: Why the name Fermat’s last theorem? Fermat proved many things but also made many conjectures which others were able to solve years later. This marginal note was the last of Fermat’s unsolved conjectures.